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IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD)

Recovering QSVT Polynomials from Side-Channel Information on Quantum Computers

Kidus Tessma, Hrvoje Kukina, Jakub M. Szefer

Published Nov 10, 2025• 1 citation
Publisher

Abstract

Quantum Singular Value Transformation (QSVT) is a powerful framework that can be applied across a wide range of quantum applications, including solving linear systems of equations, phase estimation, or amplitude amplification as employed in Grover's search algorithm. QSVT is in effect a metaalgorithm that can be used to realize various other functionalities or applications. QSVT is typically configured with different polynomials to realize the different functions, and the polynomials are in turn encoded into quantum gate operations that execute on the quantum computer. If an adversary is able to recover the gate operations, specifically gate rotations and their angles, from the quantum computer, they can then attempt to recover the polynomial used, and from the polynomial they can attempt to recover the function that the victim is executing. This paper evaluates different functions implemented in QSVT and how they map to different polynomials and the phase angles. It focuses on the correlation between the phase angle values and the number of phase angles for the different functions implemented using QSVT. The paper shows that knowing the phase angle values, or even just the number of phase angles, something an attacker can learn through a side channel, allows an attacker to learn the type of functionality being implemented by QSVT.

DOI
10.1109/iccd65941.2025.00056

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Quantum Singular Value Transformation (QSVT) is a powerful framework that can be applied across a wide range of quantum applications, including solving linear systems of equations, phase estimation, or amplitude amplification as employed in Grover's search algorithm. QSVT is in effect a metaalgorithm that can be used to realize various other functionalities or applications. QSVT is typically configured with different polynomials to realize the different functions, and the polynomials are in turn encoded into quantum gate operations that execute on the quantum computer. If an adversary is able to recover the gate operations, specifically gate rotations and their angles, from the quantum computer, they can then attempt to recover the polynomial used, and from the polynomial they can attempt to recover the function that the victim is executing. This paper evaluates different functions implemented in QSVT and how they map to different polynomials and the phase angles. It focuses on the correlation between the phase angle values and the number of phase angles for the different functions implemented using QSVT. The paper shows that knowing the phase angle values, or even just the number of phase angles, something an attacker can learn through a side channel, allows an attacker to learn the type of functionality being implemented by QSVT.
Quantum singular value transformation (QSVT) is a powerful framework that unifies many quantum algorithms, including Grover’s search, by enabling polynomial transformations of the singular values of linear operators embedded in unitary matrices. We present an improved quantum search algorithm within this framework. Instead of amplifying the target state amplitude, our method suppresses the non-target amplitude using a modified Chebyshev polynomial. This approach achieves the optimal O(Nlog⁡(1/δ))\mathcal {O}(\sqrt {N}\log (1/\delta))O(N​log(1/δ)) query complexity while requiring lower-degree polynomials. For N = 512, our method reduces the polynomial degree from 71 to 43 compared to prior QSVT-based fixed-point search. We also demonstrate how QSVT enables exact search construction.
The quantum singular value transformation (QSVT) algorithm is a general framework to implement most of the known algorithms and provides a way forward for designing new algorithms. In the present work, the impact of noise on the QSVT algorithm is examined for bit flip, phase flip, bit-phase flip, and depolarizing noise models for a small number of qubits. The small number of noisy qubits approximates the currently available noisy quantum computers. For simulation results, the QSVT implementation of the Grover search and quantum phase estimation (QPE) algorithms is considered. These algorithms are among the basic quantum algorithms and form the building blocks of various applications of quantum algorithms. The results showed that the QSVT implementation of the Grover search and QPE algorithms has a consistently worse dependence upon noise than the original implementation for all four noise models. The probability of success of the Grover algorithm and phase measured by the QPE algorithm were found to exponentially depend upon the error probability in the noisy channels but only linearly dependent on the number of qubits.
Quantum singular value transformation (QSVT) is a powerful quantum algorithm that can effectively perform a polynomial transformation of the singular values of a linear operator embedded in a unitary matrix. This article extends QSVT to perform a polynomial transformation of the amplitudes of a quantum state. This article presents an algorithm that can convert quantum state∑i=0N−1xi∣i⟩\sum_{i=0}^{N-1}x_{i}\vert i\rangle∑i=0N−1​xi​∣i⟩to quantum state1∑i=0N−1∣f(xi)∣2∑i=0N−1f(xi)∣⟩\frac{1}{\sqrt{\sum_{i=0}^{N-1}\vert f(x_{i})\vert^{2}}}\sum_{i=0}^{N-1}f(x_{i})\vert\rangle∑i=0N−1​∣f(xi​)∣2​1​∑i=0N−1​f(xi​)∣⟩and demonstrates the application of the algorithm in the quantum state small amplitude extraction.
Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.
Quantum signal processing (QSP) and the quantum singular value transformation (QSVT) are pivotal tools for simplifying the development of quantum algorithms. These techniques leverage polynomial transformations on the eigenvalues or singular values of block-encoded matrices, achieved with the use of just one control qubit. In contexts where the control qubit is used to extract information about the eigenvalues or singular values, the amount of extractable information is limited to one bit per protocol. In this work, we extend the original QSP ansatz by introducing multiple control qubits. We show that, much like in the single-qubit case, nearly any vector of polynomials can be implemented with a multi-qubit QSP ansatz, and the gate complexity scales polynomially with the dimension of such states. Moreover, assuming that powers of the matrix to transform are easily implementable - as in Shor's algorithm - we can achieve polynomial transformations with degrees that scale exponentially with the number of control qubits. This work aims to provide a partial characterization of the polynomials that can be implemented using this approach, with phase estimation schemes and discrete logarithm serving as illustrative examples.
Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.